1.1: Binary operations (2024)

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Definition:Binary operation

Let \(S\) be a non-empty set, and \( \star \) said to be a binary operation on \(S\), if \(a \star b \) is defined for all \(a,b \in S\). In other words, \( \star\) is a rule for any two elements in the set \(S\).

Example \(\PageIndex{1}\):

The following are binary operations on \(\mathbb{Z}\):

The arithmetic operations, addition \(+\), subtraction \(-\), multiplication \( \times \), and division \(\div \).
Define an operation oplus on \(\mathbb{Z}\) by \(a \oplus b =ab+a+b, \forall a,b \in\mathbb{Z}\).
Define an operation ominus on \(\mathbb{Z}\) by \(a \ominus b =ab+a-b, \forall a,b \in\mathbb{Z}\).
Define an operation otimes on \(\mathbb{Z}\) by \(a \otimes b =(a+b)(a+b), \forall a,b \in\mathbb{Z}\).
Define an operation oslash on \(\mathbb{Z}\) by \(a \oslash b =(a+b)(a-b), \forall a,b \in\mathbb{Z} \).
Define an operation min on \(\mathbb{Z}\) by \(a \vee b =\min \{a,b\}, \forall a,b \in\mathbb{Z}\).
Define an operation max on \(\mathbb{Z}\) by \(a \wedge b =\max \{a,b\}, \forall a,b \in\mathbb{Z}\).
Define an operation defect on \(\mathbb{Z}\) by \(a \ast_3 b = a+b-3, \forall a,b \in\mathbb{Z}\).

Lets explore the binary operations, before we proceed:

Example \(\PageIndex{2}\):

\(2 \oplus 3=(2)(3)+2+3=11\).
\(2 \otimes 3=(2+3)(2+3)=25\).
\(2 \oslash 3=(2+3)(2-3)=-5\).
\(2 \ominus 3=(2)(3)+2-3=5\).
\(2 \vee 3= 2\).
\(2 \wedge 3 =3\).

Exercise \(\PageIndex{2}\)

\(-2 \oplus 3\).
\(-2 \otimes 3\).
\(-2 \oslash 3\).
\(-2 \ominus 3\).
\(-2 \vee 3\).
\(-2 \wedge 3 \).

Answer: \(-5, 1,-5,-2,3\)

Properties:

Closure property

Definition: Closure

Let \(S\) be a non-empty set. A binary operation \( \star \) on \(S\) is said to be a closed binary operation on \(S\), if \(a \star b \in S, \forall a, b \in S\).

Summary of arithmetic operations and corresponding sets:

	\(+\)	\(\times\)	\(-\)	\(\div\)
\(\mathbb{Z_+}\)	closed	closed	not closed	not closed
\(\mathbb{Z}\)	closed	closed	closed	not closed
\(\mathbb{Q}\)	closed	closed	closed	closed (only when \(0\) is not included)
\(\mathbb{R}\)	closed	closed	closed	closed (only when \(0\) is not included)

Associative property

Definition: Associative

Let \(S\) be a subset of \(\mathbb{Z}\). A binary operation \( \star \) on \(S\) is said to be associative , if \( (a \star b) \star c = a \star (b \star c) , \forall a, b,c \in S\).

We shall assume the fact that the addition (\(+\)) and the multiplication (\( \times \)) are associative on \(\mathbb{Z_+}\). (You don't need to prove them!).

Below is an example of proof when the statement is True.

Example \(\PageIndex{5}\): Associative

Determine whether the binary operation oplus is associative on \(\mathbb{Z}\).

We shall show that the binary operation oplus is associative on \(\mathbb{Z}\).

Proof:

Let \(a,b,c \in \mathbb{Z}\). Then consider, \((a \oplus b) \oplus c = (ab+a+b) \oplus c = (ab+a+b)c+(ab+a+b)+c= (ab)c+ac+bc+ab+a+b+c\).

On the other hand, \(a \oplus (b \oplus c)=a \oplus (bc+b+c)= a(bc+b+c)+a+(bc+b+c)=a(bc)+ab+ac+a+bc+b+c. \)

Since multiplication is associative on \(\mathbb{Z}\), \((a \oplus b) \oplus c =a \oplus (b \oplus c). \)

Thus, the binary operation oplus is associative on \(\mathbb{Z}\). \( \Box\)

Below is an example of how to disprove when a statement is False.

Example \(\PageIndex{6}\): Not Associative

Determine whether the binary operation subtraction (\( -\)) is associative on \(\mathbb{Z}\).

Answer: The binary operation subtraction (\( -\)) is not associative on \(\mathbb{Z}\).

Counter Example:

Choose \( a=2,b=3, c=4,\) then \((2-3)-4=-1-4=-5 \), but \(2-(3-4)=2-(-1)=2+1=3\).

Hence the binary operation subtraction (\( -\)) is not associative on \(\mathbb{Z}\).

Commutative property

Definition: Commutative property

Let \(S\) be a non-empty set. A binary operation \( \star \) on \(S\) is said to be commutative, if \( a \star b = b \star a,\forall a, b \in S\).

We shall assume the fact that the addition (\(+\)) and the multiplication( \( \times \)) are commutative on \(\mathbb{Z_+}\). (You don't need to prove them!).

Below is the proof of subtraction (\( -\)) NOT being commutative.

Example \(\PageIndex{7}\): NOT Commutative

Determine whether the binary operation subtraction \( -\) is commutative on \(\mathbb{Z}\).

Counter Example:

Choose \(a=3\) and \(b=4\).

Then \(a-b=3-4=-1\), and \(b-a= 4-3=1\).

Hence the binary operation subtraction \( -\) is not commutative on \(\mathbb{Z}\).

Example \(\PageIndex{8}\): Commutative

Determine whether the binary operation oplus is commutative on \(\mathbb{Z}\).

We shall show that the binary operation oplus is commutative on \(\mathbb{Z}\).

Proof:

Let \(a,b \in \mathbb{Z}\).

Then consider, \((a \oplus b) = (ab+a+b).\)

On the other hand, \( (b \oplus a) = ba+b+a. \)

Since multiplication is associative on \(\mathbb{Z}\), \((a \oplus b) = (b \oplus a). \)

Thus, the binary operation oplus is commutative on \(\mathbb{Z}\). \( \Box\)

Identity

Definition: Identity

A non-empty set \(S\) with binary operation \( \star \), is said to have an identity \(e \in S\), if \( e \star a=a\star e=a, \forall a \in S.\)

Note that \(0\) is called additive identity on \(( \mathbb{Z}, +)\), and \(1\) is called multiplicative identity on \(( \mathbb{Z}, \times )\).

Example \(\PageIndex{9}\): Is identity unique?

Let \(S\) be a non-empty set and let \(\star\) be a binary operation on \(S\). If \(e_1\) and \(e_2\) are two identities in \((S,\star) \), then \(e_1=e_2\).

Proof:

Suppose that \(e_1\) and \(e_2\) are two identities in \((S,\star) \).

Then \(e_1=e_1 \star e_2=e_2.\)

Hence identity is unique. \( \Box\)

Example \(\PageIndex{10}\): Identity

Does \(( \mathbb{Z}, \oplus )\) have an identity?

Answer

Let \(e\) be the identity on \(( \mathbb{Z}, \oplus )\).

Then \( e \oplus a=a\oplus e=a, \forall a \in \mathbb{Z}.\)

Thus \(ea+e+a=a\), and \(ae+a+e=a\) \(\forall a \in \mathbb{Z}.\)

Since \(ea+e+a=a\) \(\forall a \in \mathbb{Z},\) \(ea+e=0 \implies e(a+1)=0\) \(\forall a \in \mathbb{Z}.\)

Therefore \(e=0\).

Now \( 0 \oplus a=a\oplus 0=a, \forall a \in \mathbb{Z}.\)

Hence \(0\) is the identity on \(( \mathbb{Z}, \oplus )\).

Example \(\PageIndex{11}\):

Does \(( \mathbb{Z}, \otimes )\) have an identity?

Answer

Let \(e\) be the identity on \(( \mathbb{Z}, \otimes )\).

Then \( e \otimes a=a \otimes e=a, \forall a \in \mathbb{Z}.\)

Thus \((e+a)(e+a)=(a+e)(a+e) =a, \forall a \in \mathbb{Z}.\)

Now, \( (a+e)(a+e) =a,\forall a \in \mathbb{Z}.\)

\(\implies a^2+2ea+e^2=a,\forall a \in \mathbb{Z}.\)

Choose \(a=0\) then \(e=0\).

If \(e=0\) then \( a^2=a,\forall a \in \mathbb{Z}.\)

This is a contradiction. Thus \(e=0\) is not an identity. Hence \(e\ne 0.\)

Choose \(a=1\). Then \(2e+e^2=0 \implese(2+e)=0.\) Since \(e\ne 0\), \(e=-2\) This will not work for \(a=0.\)

For any other values of \(e\) will not work \(a=0\).

Hence, \(( \mathbb{Z}, \otimes )\) has no identity.

Distributive Property

Definition: Distributive property

Let \(S\) be a non-empty set. Let \(\star_1\) and \( \star_2\) be two different binary operations on \(S\).

Then \(\star_1\) is said to be distributive over \( \star_2\) on \(S \) if \( a \star_1 (b \star_2 c)= (a\star_1 b) \star_2 (a \star_1 c), \forall a,b,c,\in S \).

Note that the multiplication distributes over the addition on \(\mathbb{Z}.\) That is, \(4(10+6)=(4)(10)+(4)(6)=40+24=64\).

Further, we extend to \( (a+b)(c+d) =ac+ad+bc+bd\) (FOIL).

F-First

O-Outer

I-Inner

L-Last

This property is very useful to find \((26)(27)\) as shown below:

Example \(\PageIndex{12}\): Find \((26)(27)\)

	\(20\)	\(6\)
\(20\)	\(400\)	\(120\)
\(7\)	\(140\)	\(42\)

Hence \((26)(27) =400+120+140+42=702\).

Let's play a game!

Example \(\PageIndex{13}\):

Does multiplication distribute over subtraction?

Example \(\PageIndex{14}\):

Does division distribute over addition ?

Answer

Counter Example:

Choose \(a = 2, b = 3, \) and \(c = 4.\)

Then \(a \div (b + c) = 2 \div(3+4)\)

\(= 2 \div7.\)

\(= \frac{2}{7}\).

and \( (a \divb) + (a \divc) = \frac{2}{3}+ \frac{2}{4}\).

\(= \frac{7}{6}\).

Since \(\frac{2}{7} \ne \frac{7}{6}\), the binary operation \(\div\) is not distributive over \(+.\)

Example \(\PageIndex{15}\):

Does \( \otimes\) distribute over \(\oplus\) on \(\mathbb{Z}\) ?

Answer

Counter Example:

Choose \(a = 2, b = 3, \) and \(c = 4.\)

Then \(2\otimes (3 \oplus 4) = 2\otimes [(3)(4)+3+4]\)

\(= 2\otimes 19\)

\(= (2+19)(2+19)\)

\(= 441\)

and \( (2\otimes 3)\oplus (2 \otimes 4)=[(2+3)(2+3)] \oplus [(2+4)(2+4)]\)

\(= 25 \oplus 36\)

\(= (25)(36)+25+36\)

\(= 961.\)

Since \(441 \ne961,\) the binary operation \( \otimes\) is not distributive over \(\oplus \) on \(\mathbb{Z}\).

Summary

In this section, we have learned the following for a non-empty set \(S\):

Binary operation,
Closure property,
Associative property,
Commutative property,
Distributive property, and
Identity.